


Successive Over-Relaxation

by Bitenomnom



Series: Mathematical Proof [34]
Category: Sherlock (TV)
Genre: Foot Massage, Gen, M/M, Massage, Mathematics, joint popping
Language: English
Status: Completed
Published: 2012-11-07
Updated: 2012-11-07
Packaged: 2017-11-18 04:04:22
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 1,988
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/556707
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>“Oh, give them here, you git,” is what John says to Sherlock as Sherlock rubs at his feet in the most histrionic fashion possible.<br/>	Sherlock quirks an eyebrow at John.<br/>	“Yes, right, you heard me, put your feet up here and let me rub them; you’re doing a bloody awful job of it.”</p>
            </blockquote>





	Successive Over-Relaxation

**Author's Note:**

> I had different, mathier and/or pornier plans for this one, but then they did not happen.
> 
> And yes, that described foot thing is totally possible; I do it all the time. Apparently it is some sort of crazy secret thing because it always surprises people when I do it. I can take a video or something if you need proof. XD (And: it does feel really great, especially if you've let it build up for a long time. You should try it. )  
> I guess I should also say, if joint-popping squicks you, you might not like a not inconsiderable chunk of this one -- sorry! 
> 
> If the math gods smile upon me and one day give me the appropriate lecture content for me to somehow work in a butt massage, there will so be a sequel. =p Butt massages are the best massages.

The Successive Over-Relaxation (SOR) method is used to accelerate the convergence for systems that are convergent by the Gauss-Seidel method by introducing a relaxation parameter, denoted by ω.

 

This is one method for solving a linear system of equations (which can also be written in the form of matrices) numerically—that is, say you have a set of a bunch of equations, with an equal number of unknown variables. If you don’t have too many, you can just solve using a technique called Gaussian Elimination and/or by inverting the matrix, but if you have a very, very large system, with a very sparse matrix, this can be very difficult and time-consuming. We then resort to a method where we make an initial guess at the answer and then gradually work our way toward a numerical solution to the problem. Because of the properties of the methods discussed here, any initial guess, no matter how far off, should eventually lead to the correct solution.

 

The SOR method can be derived in the following way:  
  
Ax = b (linear system of equations)

ωAx = ωb (obviously ω is not equal to zero—if it is, then we trivially get 0=0).

Qx = (Q – ωA)x + ωb (add some matrix Q multiplied to x to both sides, then subtract ωAx to the right side)

 

We define the matrix A = D – L – U (D a diagonal matrix, L is lower triangular, and U is upper triangular) and Q = D – ωL for this method (it is lower triangular). A matrix is “lower triangular” if all the entries are in the bottom-left portion of the matrix; the opposite for “upper triangular.”

 

Then, make some replacements to get

 

(D- ωL)x(k) = (D- ωL- ωA)x(k-1) \+ ωb

(D- ωL)x(k) = (D- ωL- ω(D-L-U))x(k-1) \+ ωb

(D- ωL)x(k) = ((1-ω)D + ωU)x(k-1) \+ ωb

 

For certain choices of ω in this method, we will get faster convergence than just using the Gauss-Seidel method (which would be the equivalent to choosing ω=1). There is an algorithm for solving this for each x in the x vector… (But the notation is a bit awkward to try to include here on AO3, so just let me know if you’d like to see it and I’ll whip something pretty up and link you to it.)

 

***

  
            It begins because today they did an entire chase completely on foot and all at once (no mad cab rides with Sherlock nearly breaking into the front to steal the wheel, no pausing outside a building and staking out, waiting for the suspect to leave), for an entire four miles.

            Each time they thought the culprit was going to slow down and be reasonable about it, he turned a corner and kept going. At first, they hoped he’d think he’d shaken them off and slow down, but it became clear that he was well aware that there were two very pesky and persistent men on his tail. Eventually, John had found an opening to another street where he was able to go round and cut him off on the other side of a building. Still, they only barely had enough breath to talk him down from threats of violence, and, once the police arrived, leaned back against the wall and huffed for breath, and laughed (but mostly huffed for breath).

            But that’s what they get, John supposes, for trying to chase down a marathon-runner-gone-drug-runner.

            As it turns out, Sherlock’s posh dress shoes are not ideal for running anything more than a short sprint (which, granted, is usually all he needs to do). John considers suggesting that Sherlock wear trainers, then thinks of how ridiculous he’d look in his sharp trousers and jackets and tight shirts with _trainers_ , and so considers suggesting that Sherlock wear trainers and some nice well-fitted jeans, but then thinks of…well. Not a lot, really, since at that point his head begins buzzing a little.

            “Oh, give them here, you git,” is what John says to Sherlock as Sherlock rubs at his feet in the most histrionic fashion possible.

            Sherlock quirks an eyebrow at John.

            “Yes, right, you heard me, put your feet up here and let me rub them; you’re doing a bloody awful job of it.” He nods toward Sherlock’s overdramatic motions.

            “Fine,” Sherlock finally says, and he curls his legs up and rotates about on the sofa so that he can stick his legs back out and rest his feet on John’s lap.

            “Don’t act like you weren’t hoping for me to offer.” John smirks.

            Sherlock glances to the side sheepishly, and John almost regrets saying it—perhaps this is a bit of an unusual area for Sherlock. Surely there cannot have been all that many people in the world who have offered to rub his feet; certainly, Sherlock would not think to ask it of anyone, either. (Personal comfort, thinks John, is most definitely not Sherlock’s area.)

            “I guess I should ask if you’re ticklish,” John says.

            Sherlock shrugs, still looking away.

            “You don’t know?”

            “Not exactly at the top of my priority list as experiments go.”

            “Huh.” John takes hold of Sherlock’s left foot, gripping it gently near the ankle with his right hand and slowly running his left thumb along Sherlock’s arch. “Most people learn that sort of information from irritating siblings or obnoxious friends, rather than scientific experiments, but I suppose Mycroft wasn’t the tickling type…” Sherlock wrinkles his nose. “And…” he trails off. It would be stupid to assume that however few friends Sherlock thinks he has now, he never had any before; still, though, John has a difficult time envisioning a young Sherlock leaving his feet bare for just anyone to touch.

            “I suppose you’re sufficiently obnoxious to qualify,” Sherlock finally says, quietly.

            John cannot restrain a smile. “Yeah? Is that an invitation?” He begins wiggling his fingers, ghosting over the inside of Sherlock’s arch, and Sherlock freezes, squirms, torso twisting, hips waggling, legs struggling to kick free.

            “ _John_!” he manages, and wrestles his foot out of John’s grip.

            “Hah,” John smirks. “There’s your answer: absolutely, unquestionably ticklish.” He leans back. “Now, give it back. I promise I won’t do it again.”

            Hesitantly, Sherlock stretches his legs back out, his cheeks a bright pink from the struggle. John grasps him again, and this time begins working his thumb in slow circles. “Where does it hurt the worst?” he asks after a few seconds. When he is met with silence, John looks back to Sherlock, still pink, still apparently recovering from the short tickling session, as his eyes are distant, distracted. “Sherlock?”

            “Heel,” he finally mumbles, and John swears he hears a crack in Sherlock’s voice.

            “Right,” John works his way back, pinching down with his thumb and forefinger on either side of Sherlock’s heel and then running them over his Achilles tendon. “Where else?”

            “Everywhere,” says Sherlock, still dazed, and John begins to wonder if he went too far with the tickling.

            “Are you okay?” he asks, moving back up Sherlock’s foot, pressing his thumbs deeply into the bottom of it, rubbing circles. He traces along the bones on top with his index finger, over and over and over, _one, two, three, four, five_. John pinches Sherlock’s toes, tentatively bends them forward to pop them one by one.

            After the first two, Sherlock groans.

            “Sherlock? Did that hurt? Do you want me to stop?”  
            “No,” breathes Sherlock. “Keep going.”

            And John does. He rubs gently against the bottoms of Sherlock’s toes, and then grabs each again and twists it to the side, yielding more popping sounds.

            “Does that feel okay?”

            “Not until you do my other foot.”

            John chuckles and obliges, taking the other foot into his hands. He takes his time with this one, too, staring at Sherlock as he does. Sherlock’s thrown an arm over his eyes, appears to be breathing deep, relaxed breaths. _Relaxed_ —a rare sight, for Sherlock; even when he is lounging about, he usually radiates nervous energy, _need a case, need the Work, need to use my brain_ —but not now. John meticulously pops each of Sherlock’s toes; he watches Sherlock suck in a quick gasp and then let out long, slow, controlled breaths. John finds himself timing his movements with the rise and fall of Sherlock’s chest.

            “I take it you don’t usually pop your joints,” he finally says, when Sherlock groans again at the last snap of his pinkie toe. Sherlock shakes his head. “Watch out, it gets a bit addictive. Then again, I guess if it relaxes you this much, that’s not such a bad thing.” He straightens up a bit, sets Sherlock’s foot down. “So, then, what’s your opinion? On foot massages, I mean.”

            “You’re quite good at them,” Sherlock says, and he speaks it as if something else entirely is coming out of his mouth, something desperate and lewd. John flushes a bit at the sound of it; it isn’t that it’s terribly different from how Sherlock normally sounds, it’s that it’s terribly _similar_. He’s _always_ like this, the distracting bastard, any time he speaks slowly—John feels as if Sherlock’s long fingers are plucking at his guts, with such a low frequency and such a great wavelength that it _would_ be something like cords of muscle or stretches of intestine resonating with the sound, wouldn’t it?  
            “Hold on,” John says. “I’ve got one more, if your feet can do it.”

            “My feet can do a lot of things,” Sherlock says, and John is fairly certain no innuendo is meant by it, but Sherlock has hit the resonant frequency of his innards, so he can no longer decipher much of anything but the vibrations heating up his chest. John picks Sherlock’s left foot back up and gently wraps his hand around the top, resting the heel of his palm along the outside of his foot, and twists the outside half of Sherlock’s foot inward. A muted _pop_ works its way from Sherlock’s foot to John’s hand, to John’s chest, although that may just be the way he startles at Sherlock’s low moan.

            “Other,” says Sherlock.

            John takes it and does the same, and receives another, fainter utterance from Sherlock.

            “I can show you how to do it, if you want.” Not that Sherlock couldn’t figure it out himself, John thinks.

            “No,” says Sherlock. “I’ll just let you.” John can’t hold in a chortle at the thought that Sherlock seems to view touching and holding his foot as an honor; but then, he thinks, quieting down, perhaps it is.

            “I am going to take a nap,” Sherlock declares after a moment. “Then, I shall let you massage my hands.”

            John flushes, debates pushing his luck. “And then?”

            Sherlock seems to be thinking; John hopes it’s not of the best way to tell John to piss off. Finally, he speaks: “I’m afraid this sofa is dreadful for my back.”

            John considers some remark like _no one’s making you sleep on the sofa_ or possibly _obviously_ or maybe just _no shit_ , but none of those things are conducive to John’s hands all over the lovely, creamy landscape of Sherlock’s back, so he says, “Well, I could probably help with that, too.”

            “Mm,” Sherlock says. “Good.” He closes his eyes, digs his feet under John’s thighs to warm them. John sighs and reaches for his laptop. If this is what he has to do to get Sherlock to sleep, he’s fine with it. At least this way maybe Sherlock won’t die of a heart attack in about two years, always so wired. Sherlock throws a nearby blanket over his face to block out the light coming in through the windows.

            He mumbles something nearly unintelligible through the blanket before he drifts off, and John struggles not to get his hopes up, but it _does_ sound an awful lot like, “You know what else is sore from running?

            “My arse.”


End file.
